Aircraft: B767
Phase of flight: Taxi
Anomaly: Aircraft Equipment Problem Less Severe

Shortly after completing the Parking Checklist the ground personnel via the headset informed us that the left engine was still running. This puzzled us as we had verified 'fuel flow zero' on the Parking Checklist. Closer examination indicated that the left engine was still turning albeit at a reduced RPM. My initial first reaction was that the fuel cutoff valve had failed to completely close. I had experienced this before. My next step was to pull the engine fire extinguisher handle which did not alleviate the situation. The ground person speaking to me through the headset was attempting to give me instructions which I could not initially understand. I finally understood that he wanted me to turn off the APU (Auxiliary Power Unit) Bleed valve which I did. By accomplishing this the Left Engine did spool down to 0 RPM. At this point I began to perceive that the engine was not actually running but rather was being driven by its respective starter for a then unknown reason. At some point I perceived that the ground person I was speaking with on the headset was our mechanic. When the mechanic came to the flight deck he discovered that the left engine start switch was about halfway into the ground position. It is my belief that the left engine starter engaged and began turning the engine after it was shut down. Closing the APU bleed air switch removed the air necessary to drive the starter.I cannot offer an explanation how the left engine start switch ended up in the 'half' ground position. During flight we had no indications from the overhead panel or the EICAS system that anything was amiss. It is my belief that NO limitations were exceeded as this event lasted a very short time. Upon arrival; the First Officer shut down both engines; the Parking Checklist was completed; and fuel flow was confirmed at zero. Ground personnel told the Captain the left engine appeared to still be running as passengers were deplaning. The Captain asked me to go out on the air stairs to check and it appeared to be running. He then pulled the fire handle to shut down the engine and maintenance asked him to disconnect APU (Auxiliary Power Unit) bleed air as well.Upon arrival in the cockpit; Maintenance pointed out the engine start selector was in the ground position and the start valve light was illuminated with APU bleed selected. I monitored the engine start and am 100 percent confident I heard the 'click' as the start sequence finished. We had no indications of any start valve abnormalities as well during the flight. As to not notice the start valve light when the engine shut down; I didn't look up at the switch as the Captain and First Officer were focused on what they thought was an engine not shut down correctly. I have no idea why the start selector was in the ground position and honestly did not notice it during the flight. Upon arrival; we were given several taxi instructions to the gate around aircraft pushing and I was focused on that and didn't notice the switch in the incorrect position. After engine shutdown and checklists were complete I was 'heads down' cleaning up the cockpit/stowing the clipboard/throwing out garbage. The Captain stated he thinks he may have inadvertently engaged the switch when lifting his clipboard sometime during the flight. A logbook entry was made by the Captain regarding these events. B767 flight crew reported inadvertently engaging the air start on the left engine; causing the engine to continue turning after shutdown.